Common approaches to data compression involve sampling, digitizing, and then compressing a data signal. Such approaches often include Nyquist sampling of the data signal, whereby the data signal is sampled at least twice as fast as the data signal's maximum rate of change. However, for data signals that are characterized by a high degree of data redundancy (i.e., signals exhibiting a significant degree of difference between the rate of change of a relatively sparse signal and the rate of information in it), such common approaches to data compression may be inefficient or otherwise undesirable. Examples of data signals that may exhibit high degrees of data redundancy include, but are not limited to, bio-signals, industrial signals, computer-processing signals, audio-recording signals, and video-recording signals, among others.
Compressed sensing involves an approach to sampling of a data signal that does not necessarily require sampling of the data signal at the Nyquist rate. Compressed-sensing techniques utilize the insight that a sparse signal may be sampled at a much lower rate than the Nyquist rate, and may still be accurately recovered using post-processing in hardware or software. As such, the power consumption and complexity of a signal-acquisition system may be reduced using compressed sensing.
The theoretical framework of compressed sensing is based on the matrix equation [Y]=[Φ][X]; i.e., an uncompressed input signal (vector [X]) of size N, is multiplied by a measurement matrix [Φ] of size M×N to obtain a compressed output signal (vector [Y]) of size M<N. Because [Φ] is non-invertible (i.e., not a square matrix), various algorithms may then be applied to [Y] in an attempt to search for the right [X] from several possible solutions. Such algorithms may exploit the sparsity of the data signal for high accuracy of recovery.
One possible implementation of such compressed-sensing techniques involves the multiplication of each row of [Φ] by the input signal (vector [X]). For various reasons, there are a number of limitations to this approach. Such limitations include, for example, potential inaccuracy and imprecision, power inefficiency, long delays, and large circuit areas.
An improvement is therefore desired.